SIAM Journal on Control and Optimization, Vol.49, No.2, 599-610, 2011
AN EXTENSION OF THE SCHWARZKOPF MULTIPLIER RULE IN OPTIMAL CONTROL
The search for multiplier rules in dynamic optimization has been an important theme in the subject for over a century; it was central in the classical calculus of variations, and the Pontryagin maximum principle of optimal control theory is part of this quest. A more recent thread has involved problems with so-called mixed constraints involving the control and state variables jointly, a subject which now boasts a considerable literature. Recently, Clarke and de Pinho proved a general multiplier rule for such problems that extends and subsumes rather directly most of the available results, namely, those which postulate some kind of rank condition or, more generally, a constraint qualification (or generalized Mangasarian-Fromowitz condition). An exception to this approach is due to Schwarzkopf, whose well-known theorem replaces the rank hypothesis, for relaxed problems, by one of covering. The purpose of this article is to show how to obtain this type of theorem from the general multiplier rule of Clarke and de Pinho. In so doing, we subsume, extend, and correct the currently available versions of Schwarzkopf's result.